Mathematical model

Four-state master equation (OIC + bleached)

The kinetic model augments the conformational O-I-C manifold with an absorbing bleached state B:

\[ \frac{d\mathbf{P}(t)}{dt} = Q\,\mathbf{P}(t), \quad \mathbf{P}(t)=\begin{bmatrix}P_O(t)\\P_I(t)\\P_C(t)\\P_B(t)\end{bmatrix}. \]

Rate matrix:

\[ Q= \begin{bmatrix} -(k_{OI}+k_{BO}) & k_{IO} & 0 & 0 \\ k_{OI} & -(k_{IO}+k_{IC}+k_{BI}) & k_{CI} & 0 \\ 0 & k_{IC} & -(k_{CI}+k_{BC}) & 0 \\ k_{BO} & k_{BI} & k_{BC} & 0 \end{bmatrix}. \]

The seven kinetic rates are:

• k_OI: Open → Intermediate
• k_IO: Intermediate → Open
• k_IC: Intermediate → Closed
• k_CI: Closed → Intermediate
• k_BO: Open → Bleached
• k_BI: Intermediate → Bleached
• k_BC: Closed → Bleached

Emission model in FRET space

Conformational FRET levels are:

• E_O, E_I, E_C for open/intermediate/closed states.

Initial conditions are parameterized by two probabilities:

• π_O and π_I (with π_C = 1 - π_O - π_I, π_B = 0).

Dynamic signal:

\[ \mathbf{P}(t) = e^{Qt}\,\pi, \quad E_{\text{dyn}}(t) = \begin{bmatrix}E_O & E_I & E_C\end{bmatrix} \begin{bmatrix}P_O(t)\\P_I(t)\\P_C(t)\end{bmatrix}. \]

Total ensemble signal:

\[ E_{\text{total}}(t)=f_{\text{dyn}}\,E_{\text{dyn}}(t)+(1-f_{\text{dyn}})\,E_{\text{static}}. \]

Likelihood / goodness-of-fit

Given observed mean trajectory \hat E(t_i) with optional uncertainty weights w_i, fitting minimizes a weighted residual objective,

\[ \mathcal{L}(\theta) \propto \sum_i w_i\left(\hat E(t_i)-E_{\text{total}}(t_i;\theta)\right)^2, \]

and reports diagnostic metrics (RMSE, residual structure, and bootstrap intervals) for parameter reliability.

Numerical stability

Matrix exponentials can be sensitive when rates are near-degenerate or eigenvalues are near zero. Constrain rates to physically plausible bounds and validate solutions with residual diagnostics.

Bootstrap interpretation

Interpret bootstrap intervals on the 7 rates as empirical uncertainty under trajectory resampling. Narrow intervals indicate robust identifiability; broad or skewed intervals suggest parameter coupling or weakly informative data.